The central limit theorem for Markov chains started at a point. (English) Zbl 1012.60028

Summary: The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the \(L^2\)-norms of the ergodic sums for the function generating the additive functional, which must be \(O(n^\alpha)\) with \(\alpha<\frac 12\). The result holds almost surely with respect to the invariant probability of the chain.


60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
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